On strong sharp phase transition in the random connection model

We consider a random connection model (RCM) $ξ$ driven by a Poisson process $η$. We derive exponential moment bounds for an arbitrary cluster, provided that the intensity $t$ of $η$ is below a certain critical intensity $t_T$. The associated subcritical regime is characterized by a finite mean cluster size, uniformly in space. Under an exponential decay assumption on the connection function, we also show that the cluster diameters are exponentially small as well. In the important stationary marked case and under a uniform moment bound on the connection function, we show that $t_T$ coincides with $t_c$, the largest $t$ for which $ξ$ does not percolate. In this case, we also derive some percolation mean field bounds. These findings generalize some of the recent results. Even in the classical unmarked case, our results are more general than what has been previously known. Our proofs are partially based on some stochastic monotonicity properties, which might be of interest in their own right.

💡 Research Summary

This paper investigates the percolation phase transition of the random connection model (RCM) driven by a Poisson point process. The authors introduce two critical intensity parameters: t_c, the classical percolation threshold beyond which an infinite cluster appears, and t_T, the supremum of intensities for which the expected cluster size remains uniformly bounded in space. They prove that for any intensity t < t_T the cluster containing a typical point v satisfies a uniform exponential moment bound: there exists δ > 0 such that ess sup_v E_t exp(δ|C_v|) < ∞. This result holds under the sole assumption that the connection function φ is integrable (∫φ(x,y)λ(dy) < ∞), removing earlier monotonicity or bounded‑support requirements.

When φ additionally decays exponentially with distance, the authors show that the cluster diameter also has an exponential tail, establishing spatial confinement of subcritical clusters. The analysis relies on a continuous‑space version of the Mecke equation to express cluster growth recursively, and on spatial Markov properties together with stochastic monotonicity to control the addition of new points.

The paper then focuses on the stationary marked RCM, where points carry marks from a probability space (M,Q) and the connection function is translation‑invariant in the spatial coordinate. Under a uniform second‑moment condition on the averaged connection strength dφ(p,q)=∫φ(0,p,x−q)dx (essential supremum finite with respect to Q), the authors prove that t_T coincides with t_c. Consequently, the model exhibits a “strong sharp phase transition”: the subcritical regime (t < t_c) is characterized by uniformly exponential decay of cluster size and diameter, while the supercritical regime (t > t_c) admits a unique infinite cluster.

Further, the authors derive mean‑field type lower bounds. In the subcritical phase they obtain a lower bound on the mean cluster size χ(t)=E_t|C_0| of order (t_c−t)^{−1}, and in the supercritical phase they prove a linear lower bound on the percolation probability θ(t)=P_t(|C_0|=∞) of order (t−t_c). These bounds match classical predictions from percolation theory and extend them to the continuous‑space, possibly marked, setting.

Methodologically, the work combines the Mecke equation, a spatial Markov property (originally from